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in two diagonal planes docs not present itself in the four groups of 

 principal sections of C la . 



B. The spatial sections through an edge of C'« 00 . 



8. Through any edge AB (fig. 3) of G'„ on pass live limiting 

 tetrahedra of this cell; the five edges opposite to AB of these letra- 

 hedra are the sides of a regular pentagon l\P. A ■ • . P h , the vertices 

 of which are at the same time adjacent to A and B. In other words: 

 the icosahedra adjacent to .1 and />', situated in different spaces, 

 penetrate one another in the regular pentagon l\l\ . . . P., adjacent 

 to AB, the vertices of which are common to both. So through the 

 edge AB pass five diagonal spaces corresponding respectively to the 

 live vertices /'.. 1\. . ., J\ ; they intersect the plane of the pentagon, 

 perfectly normal in its centre M to the plane ABM, in the diagonals 

 P P , P,F„ . . . PJ\ of the pentagon, or - - if one likes — in the 

 sides of the starpentagon l\l\P,P.J\. In the case of C lt the centre 

 of the square l\PJ\l\ was at the same time the centre of* the 

 cell. Here the centre M of' the pentagon is not even the centre of 

 the two icosahedra penetrating one another, and still less the centre 

 of C too ; here the line joining M to the midpoint M' of the edge 

 Al> must contain the centre of C too . 



If the trace / of the intersecting space on the plane of the penta- 

 gon adjacent to AB cuts /'./'., in 5, fig. •">), ABS X is a diagonal 

 plane. Kor this plane is the intersection of the intersecting space 

 determined by AB and /with the diagonal space determined by AB 

 and PJ\ of the icosahedron adjacent to?,, and 5, lies on 1\1\ 

 itself', not on its production. Indeed it is evident that this icosahe- 

 dron is cut by any plane through AB and a point of J\/\, if this 

 point lies on I\I\ itself, whilst the plane will contain of this icosahe- 

 dron the edge AB only, if this point lies on P, /'., produced. In order 

 to prove this we have only to observe that the lines AB and PJ\, 

 the first of which is an edge of C 600 and the latter a chord, 

 cross one another normally. From this it ensues that these lines, 

 likewise edge and chord of the icosahedron determined by the 

 points A, II, /',, /',, can he represented (tig. 4), in projection on a 

 plane through two opposite edges pr,p'r' of the icosahedron, by 

 the edge in </ normal to the plane of the diagram and the chord 

 np' situated in that plane, the extremities of the edge being joined 

 by edges to the extremities p,p' of that chord. This shows immediately 

 that any plane through the edge projecting itself in q cuts the 

 icosahedron or not, according to whether the point of intersection 

 of the plane with pp' lies on this line itself or on its production. 



